Non-Commutative Worlds - Classical Constraints, Relativity and the Bianchi Identity

Details Non-Commutative Worlds - Classical Constraints, Relativity and the Bianchi Identity Non-Commutative Worlds - Classical Constraints, Relativity and the Bianchi Identity

2011-09-06

arxiv.org/abs/1109.1085v1

Physics

Mathematical Physics

LaTeX document, 34 pages, 7 figures

Scientific paper

This paper shows how discrete measurement leads to commutators and how discrete derivatives are naturally represented by commutators in a non-commutative extension of the calculus in which they originally occurred. We show how the square root of minus one (i) arises naturally as a time-sensitive observable for an elementary oscillator. In this sense the square root of minus one is a clock and/or a clock/observer. This sheds new light on Wick rotation, which replaces t (temporal quantity) by it. In this view, the Wick rotation replaces numerical time with elementary temporal observation. The relationship of this remark with the Heisenberg commutator [P,Q]=ihbar is explained in the Introduction. After a review of previous work, the paper begins with a section of iterants - a generalization of the complex numbers as described above. This generalization includes all of matrix algebra in a temporal interpretation. We then give a generalization of the Feynman-Dyson derivation of electromagnetism in the context of non-commutative worlds. This generalization depends upon the definitions of derivatives via commutators and upon the way the non-commutative calculus mimics standard calculus. We then begin a project of examining constraints that link standard and non-commutative calculus, summarizing work Anthony Deakin and formulating problems related to the algebra of constraints. The paper ends with a discussion of the Bianchi identity in non-commutative worlds and with an appendix about the constraint algebra.

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